Optimal. Leaf size=21 \[ \frac {2}{3} \left (1-x^2\right )^{3/2}-x^2 \]
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Rubi [A] time = 0.11, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6688, 6742, 261} \[ \frac {2}{3} \left (1-x^2\right )^{3/2}-x^2 \]
Antiderivative was successfully verified.
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Rule 261
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int x \left (-\sqrt {1-x}-\sqrt {1+x}\right ) \left (\sqrt {1-x}+\sqrt {1+x}\right ) \, dx &=-\int x \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx\\ &=-\int \left (2 x+2 x \sqrt {1-x^2}\right ) \, dx\\ &=-x^2-2 \int x \sqrt {1-x^2} \, dx\\ &=-x^2+\frac {2}{3} \left (1-x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 21, normalized size = 1.00 \[ \frac {2}{3} \left (1-x^2\right )^{3/2}-x^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 25, normalized size = 1.19 \[ -x^{2} - \frac {2}{3} \, {\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 54, normalized size = 2.57 \[ -{\left (x + 1\right )}^{2} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + 2 \, x + 2 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 1.24 \[ -x^{2}-\frac {2 \sqrt {x +1}\, \sqrt {-x +1}\, \left (x^{2}-1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 17, normalized size = 0.81 \[ -x^{2} + \frac {2}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.06, size = 25, normalized size = 1.19 \[ -x^2-\frac {2\,\left (x^2-1\right )\,\sqrt {1-x}\,\sqrt {x+1}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 101.26, size = 110, normalized size = 5.24 \[ \frac {x^{3}}{3} + x - \frac {\left (x + 1\right )^{3}}{3} + 4 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) - 4 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} - \frac {\left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{6} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) + 1 \]
Verification of antiderivative is not currently implemented for this CAS.
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